# What is the lowest GPA ever recorded

## Breakdown of the Migdal approximation at Lifshitz junctions with gigantic zero point movement in the H3S superconductor

### subjects

- Materials science
- Superconducting properties and materials

### abstract

While the high temperature superconductivity of 203 K in H _{3} S interpreted by BCS theory at the dirty limit, we will focus here on the effects of hydrogen zero motion and the electronic multiband structure relevant to multigap superconductivity near Lifshitz junctions. We describe how the topology of the Fermi surfaces develops under pressure, which leads to different Lifshitz transitions. A breakneck Lifshitz transition (type 2) occurs when the Van Hove singularity vHs crosses the chemical potential at 210 GPa and new small 2D Fermi surface areas appear with a slow Fermi velocity, for which the Migdal approximation becomes questionable . We show that the neglected hydrogen zero point movement ZPM plays a key role in Lifshitz transitions. It induces an energy shift of approximately 600 meV in the vHs. The other Lifshitz transition (of type 1) for the appearance of a new Fermi surface occurs at 130 GPa, where new Fermi surfaces appear at the Γ point of the Brillouin zone, the Migdal approximation collapsing and the zero point motion induces large ones Fluctuations. The maximum T _{ c } = 203 K occurs at 160 GPa, where E _{ F. } / ω _{0} = 1 is in the small Fermi surface pocket at at. A Feshbach-like resonance between a possible BEC-BCS condensate at Γ and the BCS condensate at different k-space locations is proposed.

### introduction

The recent discovery of superconductivity in pressurized sulfur hydride metal with a critical temperature of T _{ c } over 200 K ^{1, 2} has shown experimentally that the coherent quantum macroscopic superconducting phase can occur at a temperature higher than the lowest temperature ever measured on earth (-89.2 ° C) ^{3, 4} . High temperature superconductivity was found in other hydrides ^{5} like pressurized PH _{3} with T _{ c } at about 100 K. ^{6 found} and in other hydrides such as yttrium hydride ^{7} predicted. Superconductivity in pressurized hydrides was proposed by Ashchroft and his co-workers 8, 9. The Universal Structure Predictor available today: Evolutionary Xtallography, USPEX, Code ^{10, } allows the structure and high temperature superconductivity of pressurized sulfur hydride to be predicted ^{11, 12, 13} . The prediction of metallic H _{3} S with the

The lattice symmetry ^{12, 13} has now been determined by X-ray diffraction experiments ^{2} confirmed above 120 GPa, while different stoichiometry and structure were found at lower pressure ^{14, 15} . The low mass of the H atom drives the sulfur-hydrogen T _{2 u } - stretching vibration and the T _{1 u } Phonons at Γ too high energy, so that the energy cutoff for the pairing interaction ω _{0} = 150 ± 50 meV. The superconducting temperature was predicted by using the Allen-Dynes modified McMillan's formula ^{11, 12, 13} and the Migdal Eliashberg formula ^{16, 17, 18, 19.} Most of this work used Eliashberg's theory with BCS approximations for isotropic pairing in a single-band metal, with the fouling limit reducing a multiband to an effective single-band metal and the Migdal approximation ω _{0} / E _{ F. } << 1 were accepted. more advanced theories have density functional theory including calculations of effective Coulomb repulsion ^{20, 21} used. Within the Eliashberg theory, the Migdal approximation assumes that the electronic and ionic degrees of freedom can be strictly separated with the Born-Oppenheimer approximation, which is valid for metals where the chemical potential is far from the band edges. The division of the Migdal approach was in the multiple gap superconductor Al _{ x } Mg _{1} - _{ x } B. _{2}^{22, 23, 24, 25}where the band edge of the band σ fluctuates over the chemical potential due to zero point movement ^{26} observed. The collapse of the Migdal approximation in a multigap superconductor is relevant as it requires the correction of the chemical potential caused by pairing below the critical temperature ^{25} which is ignored in the standard Eliashberg theory. Furthermore, where the Migdal approach breaks into a multi-spacing superconductor it enetrs in an unconventional superconducting phase where the coexisting more condensates could be either in the Bose-Einstein condensate (BEC) regime or in the BEC-BCS crossover regime. In this complex more gaps superconductors where BEC, BEC-BCS and BCS - condensates the exchange interaction between the various condensates ^{25, 27, 28, 29} coexist, which is neglected in the BCS approximations, but a relevant term could be both to increase or to stabilize too high temperature superconductivity. This quantum term is a contact interaction that is given by the quantum overlap between the condensates and the condensation energy ^{25, 27, 28, 29, 30, 31} and the critical temperature about the ^{Shape resonance} analogous to the Fano-Feshbach resonance in ^{Ultra cold} increases gases. It is found that the critical temperature around the maximum gain shows where one of the condensates is in the BCS-BEC crossover that occurs on the verge of a Lifshitz transition ^{3, 4, 30, 31, 32, 33, 34, 35} . At the Lifshitz junction, a change in the topology of the Fermi surface is induced by pressure or doping, and it has been shown to reduce the high temperature superconductivity in iron pnictides ^{30, } 31 ^{controls} .

Since H _{3} S is a multiband metal and lifshitz transitions by increasing the pressure ^{3}, ^{4} could occur, it has been suggested that there is a superconducting multigap near the Lifshitz junctions ^{27}, ^{28}, ^{29 is}, where also a phase separation ^{32} in the multi-scale area at the Lifshitz transition ^{is present} might appear similar to what is in the cuprates ^{34, 35, 36, 37} observed.

The coupling between the electronic and the atomic lattice degrees of freedom in sulfur hydrides at zero temperature was neglected in earlier calculations of the electronic structure under the assumption of a very large Fermi energy. On the contrary: in the case of band edges close to the chemical potential and in the vicinity of Lifshitz electronic topological transitions of the zero point movement (ZPM) cannot be neglected. It is known that the ZPM modifies the band structure itself ^{38, 39, 40, 41, 42, 43, 44} . With corrections for the band gap energy, which can be greater than that induced by correlation. In addition, the lattice perturbation from ZPM will also cause band broadening as it is ^{41} was proven in early studies. It has been shown that such effects are important for many properties in different materials, even if their atomic masses are greater than ^{42, 43, 44} and the lattice perturbation can perturb spin waves and phonons ^{45} . It is expected that the zero point movement in H _{3} S due to the low mass of H atoms, the high-frequency SH stretching modes and the double trough potential for hydrogen in H _{3} S. ^{3} large, similar to the well-studied cases of ice, and biological macromolecules.

In this thesis we discuss the effect of lattice compression and zero point movement on the electronic structure of H. _{3} S in the pressure range above 120 GPa, where the metallic

^{27},

^{28},

^{29},

^{30},

^{31}for a Lifshitz transition towards zero it is of type 1. In addition, the maximum critical temperature 203 K appears, at which the Fermi energy in the small Fermi hole pocket at Γ is of the order of magnitude of the pairing interaction ω

_{0 is }as in Ref.

_{1}predicted. 27, 28, 29, 30, 31.

### The band structure

The high pressure phase of metallic H _{3} S has the cubic space group: 229 with

Lattice symmetry. The

The lattice structure can be described by the small Body Centered Cubic (bcc) unit cell that was used in all previous calculations to provide the electronic band dispersion in the bcc-Brillouin zone (bcc BZ).Here we also use an alternative simple cubic unit cell of 8 atoms per unit cell, the one with a simple cubic Brillouin zone (sc BZ), which allows an easier comparison with a conventional group of superconductors, namely compounds A15. In fact, A15 compounds have a lattice structure with the cubic space group: 223, belonging to the same ditesseral middle class or to the same galenic type of cubic space groups with the same Hermann-Mauguin point group

.We have obtained the band dispersion in the large simple cubic (sc) Brillouin zone, which picks up further details of the electronic band dispersion in the complex bitruncated cubic honeycomb lattice. Here S-sites form a bcc grid (just like Si in the A15 connection V _{3} Si) with linear chains of H at the boundaries of the sc cell, similar to how the transition metals in A15 superconductors form linear chains. These results allow easier comparison of electronic tapes between the two types of material.

We show the electronically self-consistent paramagnetic bands for the bcc BZ in Fig. 1 and in Fig. 2 for the sc BZ. In this last picture, the simple cubic unit cell contains a total of 8 digits. The calculations were made using the linear muffin tin orbital method (LMTO method) ^{46} and the local spin density approximation (LSDA) as part of the standard methods ^{47}, ^{48}, ^{49}, ^{50 carried out} . The basic rate goes through

_{ F) }is. The Wigner-Seitz (WS) 0.38 radii are for an S and 0.278 A for H. The K point - mesh corresponds to 11 points between Γ and X, or 1331 points completely in 1/8 of the BZ or finer K-point for plots of the bands interlocking along lines of symmetry. A spin polarized calculation is made for the largest volume from an imposed ferromagnetic (FM) configuration. All local FM moments converge towards zero, which shows that FM and FM spin fluctuations are unlikely. The deep s-band on S is the Si-s-band in the A15 connection V.

_{3}Si

^{47}very similar, but the high pressure in H

_{3}S causes it to overlap with the Sp band. In contrast, the Si-p band is in V

_{3}Si separated from the Si-s band. The separation is not complete in H

_{3}S, but it is visible as a jump in the DOS 13 eV below

_{ F. }E for the largest lattice constants, see Fig. 3.

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An approximation for the pressure P calculated as a surface integral using the virial theorem ^{51}, is useful for gaining insight into the relative contribution of different atoms. These total pressures (P) are 0.3 Mbar for the two extreme lattice constants. The partial S and H pressure increases much more at H than at S when the lattice constant is decreased. This along with the charge transfers indicate that the S sublattice is more compressible than that of H. A charge transfer from S to H at high pressure will force the hardening of H. More precise values of P shown in Table 1 will force from the volume derivative of the calculated total energies we get 180 GPa for a = 5, 5 au and 73 GPa for a = 6, 1 au, which agrees well with ref. 16.

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The NI electronic counter ^{2} of the electron-phonon coupling constant λ = NI ^{2} / K (where the force constant K = M? ^{2} is removed from the experiment) in the Rigid Muffin-Tin Approximation (RMTA) ^{52, 53} calculated. The matrix elements for the d - f - scattering in S and the p - d - scattering in H are missing, which leads to a certain underestimation of the total value of λ. It should also be noted that LMTO overlapping Wigner-Seitz (WS) spheres do not contribute to I. ^{2} used from the interstitial area. The values of NI ^{2} at H and S are relatively large compared to the values in transition metals. The low local DOS is replaced by a larger I. ^{2} and compensates for the dispersion in p-states, while in the A15 a large dispersion results mainly from d-states. Our N (E _{ F), } shown in Table 1, are in accordance with the work of Papaconstantopoulos et al. ^{16}which are unusually largest at low volume and a large P. This is because the belt edges that E _{ F. } cross and be occupied at high P. In the case of an even larger volume, N (E _{ F. } ) but returns to normal and increases as P decreases, as Papaconstantopoulos et al. ^{16} . The matrix elements I ^{2} show strong P-dependencies, and NI ^{2} (the so-called Hopfield parameter) increases steadily with P in accordance with ref. 16. The absolute value of λ is smaller than in Ref. 16. This can partly be due to the smaller basis in our case, and also because of the use of WS spheres in LMTO instead of non-overlapping MT spherical geometry in LAPW ^{16.}

In the estimate of the T _{ c } from the McMillan equation, another uncertainty concerns the Coulomb repulsion μ ^{*.} Many T _{ C. } -Calculations use μ ^{*} = 0, 11-0, 13, but theories for calculating μ ^{*} are approximate or unreliable ^{54} . The delay can be μ ^{*} large and make the shield small or even negative, depending on the bandwidth and phonon frequency ^{54} . Here we use an empirical formula for μ ^{*, } that of Bennemann-Garland ^{55 proposed} and results in a small value on the order of 0.03. We therefore warn that our calculated T _{ c } according to McMillan's formula, even if it should be large, is very approximate, similar to the other works. Our T _{ c } are on the order of 145 and 75 K between the two extreme lattice constants when the total coupling strength is between 1.0 and 0.7. This T _{ C. } ‚S are reduced to 95 and 40 K between the extreme values of a, if μ ^{*} 0.13 is. The different contributions to the total λ of each S and H atom are comparable. In addition, the sum λ itself is not unusually large. Using McMillan's formula, which is only valid for a single effective band BCS system in the dirty limit, the high T is _{ c } mainly on the large phonon frequency prefactor (ω _{ log } ) in the equation.

In order to examine the pressure effects on the electronic structure, we show in Fig. 3 the different total DOS for different lattice constant a of the perovskite structure 6, 2-5, 6 au total DOS at E _{ F. } per S-atom change is about 3, 5 (Ry) ^{−1} and 1 (Ry) ^{−1} per H-site, compared to the order of 20 per V in V _{3} Si or in elemental V and Nb. This is not surprising as the width of the wide band in H _{3} S at the bottom is 2 Ry below the chemical potential (see Fig. 3), while to increase the number of (d-) electrons in the transition metal A15, the total amount is made up. The bandwidth is more like 2/3 of a Ry ^{47} .

The charge inside the H-WS sphere increases 1, 3-1, 4 el./H, as the lattice constant decreases 6, 2-5, 6 Au, and the charge Hs goes 0.95-1.0. This fact somewhat justifies the use of LSDA for H, although atomic H with exactly 1 electron is best described only by the Hartree potential. The results show a very large effect of the pressure on the lowest dispersive bands with H s character, and the energy shift of the narrow peak of the DOS at the chemical potential due to the van Hove singularity.The shift is small in the figure because of the large energy scale.

### Hydrogen Zero Point Movement Effects

Usually the atomic velocities (v _{ i) } The vibrations are much slower than the electronic speeds, and the electronic structure can relax adiabatically at all times. Therefore, the calculations of the electronic structure can usually v _{ i } neglect, as is the case with frozen phonon calculations. However, it is known that the lattice of real materials is distorted at a large T ("thermal disturbance") and a certain distortion remains at T = 0 due to the "zero point movement", ZPM. Thermal Disorder and ZPM change and broaden the bands compared to a perfect lattice, since the potentials are not exactly the same at different points. Every atom (i) lattice position deviates from its mean position of u (t, T), due to the excitation of phonons, temporal mean values of u (T) for harmonic uncorrelated oscillations are also represented by a Gaussian distribution function with a width (FWHM) < u> what tends to

at low T (ZPM) and up to 3 kB T / K at high T, see Ref. 56.

The Debye temperature for H phonons (~ 1500 K) in sulfur hydride is much higher than T. _{ c } and the noise amplitude of ZPM is almost the same as that of T = T _{ c } . The maximum superconducting gap (2Δ) would be ~ 65 meV at T = 0 in the large Fermi surface, i.e. the band No. 2 to identify the designations with the 5 bands introduced in Ref crossing the Fermi level. 3. With the parameters as in the calculation of λ, and Δ = V, the amplitude for the potential modulation for phonons ^{57, } one can estimate that U _{Δ, } where the shift of phonons that should be leading to superconductivity is ~ 0.02 Å, i.e. smaller than the amplitude of ZPM for H or comparable to the ZPM of S.

We used a first principle DFT approach based on a large supercell to calculate the effect of ZPM on the Van Hove singularity. This method is more suitable for intricate 3D materials like H. _{3} S (a unit cell of 64 atoms) compared with alternative perturbative approaches to evaluating band energy changes as a function of displacement (u) based on the Allen-Heine-Cardona (AHC) method used for simple systems, such as carbon nanotubes ^{38, 39} suitable is. While AHC Art's approach seems to be more sophisticated when applied to complex 3-D materials, it is even more important to look at good statistics as well ^{42, 43, 44, 45} rely on experimental evidence on force constants. The important point of our approach is that in order to have good statistics for the individual u of each atom, a large super cell is used so that the bands of energy changes can be properly determined as a function of increasing perturbation. The average atomic displacements and their 3-D distribution are determined from the phonon spectrum and atomic masses, and it makes no sense to attempt an ab initio computation of u using frozen zone boundary phonons and tight-binding method.

Calculations for the perturbation in supercells with 64 atoms for FeSi ^{42} or even less (48 atom) for purple bronze ^{43, } have shown that different generations of internal disturbance in the cells produce similar results (as long as the disturbance amplitude __ are the same). Symmetry makes the bands identical in different irreducible Brillouin Zones (IBZ) for ordered supercells, and these bands have their exact correspondence in the bands of the small cell. But this symmetry is lost when the atoms are disordered, and the bands have to be determined in half of the BZ.__

__With a force constant K = Mω ^{2} of 7 eV / Å ^{2 received} we have a mean amplitude on the order of 0.15 Å for ZPM. This is close to 10 percent of the HH distance, which, according to the Lindemann criterion, suggests that the H sublattice is close to ^{56} Melt. The amplitude for the S sublattice is normal because of its large mass, and it is likely that the rigid S lattices are important for the stability of the structure in which the H atoms are rather loosely attached to their ideal ones Positions.__

__The ZPM generates different shifts in the Madelung potential at different locations. This changes the band energies ( __

__) And leads to “fluctuations” (there is a scattering of the eigenvalues in different IBZ), especially at the band edges. The low energy band T fluctuations in materials with narrower bandwidths are known to be about 20 meV for u in the range of 0.03-0.04 Å__^{42, 43, 44, 45}be. From an extrapolation of these values to the conditions in H_{3}S (greater than__) we estimate that the band energy swing will be greater than 150 meV for H bands.__The band No. 2 by H _{3} S in Fig. 3 is very wide, approx. 2 Ry for the high lying valence band and the Ss band overlaps with the Sp band. This makes the band dispersion and Fermi velocities high in large parts of k-space, in which the effect of the energy band broadening at these points of k-space is less important. However, in parts of the k-space around the-XM path in the sc BZ, in which the Van Hove singularity crosses the chemical potential, the Fermi velocity becomes small and it is expected that strong dynamic fluctuations through the zero point -Lattice fluctuations controlled are relevant.

The electronic calculations for the ZPM in H _{3} S at P = 210 GPa, a = 5, 6 au, a large supercell was carried out in which the lattice is disordered. Randomized shifts and the like are assigned to each atom _{ x }, u _{ y }, u _{ z } assigned in such a way that the distribution of all displacement amplitudes (| u |) has a bell-shaped (Gaussian) distribution with the FWHM width equal to the averaged displacement amplitude (__) on before and in Ref. 42. In Fig. 4 we show the energy renormalization of the DOS based on the calculated zero point movement (ZPM). Here we consider a 2 × 2 × 2 extension of the cubic unit cell with 64 atoms, which allows to fully calculate U of 192 displacement vectors, which is a reasonably good statistic for the calculation . As discussed above, it is sufficient to perform the calculation for a disordered configuration when the supercells are at least 48 atoms in size. Because of the large mass difference between S and H, we leave here for H (u _{ H } ) to a larger than for S (u _{ S. } ). For small it can be that the correlation of vibrational movements and anharmonic terms are small ^{56} to be shown. But u _{ H } is large and in creating a disorderly configuration for u _{ H } / a = 0.05, as for the expected ZPM, several pairs of H come too close. Therefore, in order to avoid large anharmonic effects in this phase, we have to calculate the electronic structure for a supercell with U _{ S. } / A = 0.01 (which is inferred from the expected ZPM for S) and U _{ H } / a = 0.033 (the 2/3 of the expected ZPM for H).__

__The DOS for H _{3} S for 64-site cubic supercells. The (blue) thin line is the DOS for the perfectly ordered super cell. The (red) thick line shows the DOS for the disordered supercell with U _{(S)} = 0, 01A and u _{(H)} = 0.033a zero point movement.__

__Full size image__

__The resulting DOS for the disordered grids is shown in FIG. The results show a strong effect of ZPM on the energy of the narrow tip of the DOS just below the chemical potential. This peak has a large proportion of H s and p states, while the peak continues downwards (3-4 eV below E. _{ F) } less H character and less affected by disorder. The width of the DOS peak with a large character of the H states becomes wider in the expected disordered case, but we also see a large energetic renormalization of this narrow peak. This peak, which was mostly below the chemical potential in the ordered lattice, is now pushed beyond the chemical potential due to the lattice ZPM. The shift at the top of the peak is on the order of 600 meV (see Figure 4) and the peak becomes wider in the disordered case.__

__Our simple approach to treating the ZPM is based on the assumption that the frozen perturbation in our supercell calculation is representative of the ZPM. If so, one can ask how much ZPM would change in the RMTA value of λ and the McMillan estimate of T _{ c. } The overall DOS at E _{ F. } is almost the same (7.3 and 7.4 states / eV / cell to the ordered and disordered case respectively) as can be seen in Fig. 4. The λ s and __

__for ordered and disordered cases, 0, 88 and 0, 86 and 122 K and 118 K are calculated. Thus there is a small decrease in & lgr; and T___{ c due }a fault, even if the two N (E_{ F. }) are comparable. However, one can determine that the peak in the DOS has shifted from below the chemical potential in the ordered case to above, if ZPM is taken into account, which indicates a 600 meV energy shift of the Van Hove singularity with zero-point lattice fluctuations where H is involved atoms. In the remaining sections some details about the Lifshitz transitions as a function of pressure are discussed.### Lifshitz Transitions as a function of pressure

The energy shift of the Van Hove singularity (vHz) can be followed by considering the shift of the narrow peak in the entire DOS near the chemical potential at various lattice parameters a, as shown in FIG.

For reasons of visibility, each DOS curve is around 0.5 (cell eV) ^{−1} Units separated.

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This peak is largely due to the sulfur orbital contributions, as the partial DOS functions of the vHs show for the case of the lattice parameter a = 5, 6 a. u. shown in Fig. 6. The Sp orbital and Sd orbital contribute about 30 and 16 percent of the total DOS, respectively, therefore the VHS is primarily due to sulfur. Both the total charge and the

-Character at E_{ F. }on H there are mostly s (60-65 percent), and the hybridization between the s electrons and states on the S atom away from the chains is large. This large on-site hybridization is favorable for large dipole matrix element contributions to the electron-phonon coupling. Fig. 7 shows the band breakdown of the entire DOS in the unit cell with 8 cubic sites in the sc BZ into 4 bands classified as No. 8, No. 9, No. 10, No. 11 DOS is only through 10 to band , which gives the largest Fermi surface area, and there is a van Hove singularity to be given due to the flat dispersing bands at low speed near Fermi X and M points. Figure 8 shows the pressure dependence of the vhs in band 10, which goes through the chemical potential at around 210 GPa, where a roughly 5.6 au.This result shows that the van Hove singularity reaches the chemical potential at 210 GPa, but it remains near the chemical potential in the energy range of the energy cutoff the pairing interaction in the pressure range shows high temperature superconductivity. When the vHs exceed the chemical potential, a breakneck Lifshitz type 2 transition occurs. Here the topology of the large Fermi surface changes because of the appearance of small pieces of tubular 2D Fermi surfaces connecting the large petals, as in Ref.. 3. The Fermi velocity is low in these tubular sections, which is why the Migdal approximation collapses. In contrast, in the large petals the Fermi energy is much larger and the Migdal approximation is valid.

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Only the bands 9 and 10, represented by thin and heavy solid lines, contribute significantly to the total DOS at the chemical potential E.

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In order to link the transition of the chemical potential of the narrow peak in DOS with the Lifshitz transitions on the topology of the Fermi surfaces, we recorded the electronic bands in a narrow energy range near the chemical potential for both sc BZ and bcc BZ in Figures 9 and 10. These volume plots show that multiple Lifshitz transitions appear for increasing pressure. From Fig. 9 it is that a local band maximum crosses at about 2/3 of the Γ in the sc BZ E _{ F. } seen - M distance if a ~ 5.8. Au (see Figures 1 and 2) and the same band crosses the chemical potential between N and H in Figure 10 in the bands for the bcc BZ. This Lifshitz transition occurs at P = 210 GPa. The energy difference E2 between the chemical potential and this local band maximum, which is associated with vHs, goes from –200 to +100 meV when a decreases from 6.2 to 5.6 au. This leads to a throat which disrupts the Lifshitz junction in the Fermi surface at 210 GPa the throat disappears at low pressure in the pipe ^{20} NH. The Fermi surface neck appears at the point where the narrow peaks in the DOS cross the chemical potential. Another band can be seen approaching the chemical potential in Fig. 9 between X and M as the pressure increases. However, crossing this potential band (which cannot be seen on a line of symmetry in the band plots for the small bcc cell) does not E _{ F. } reach to make an FS pocket unless P is increased further.

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The colors of the bands correspond to different grid parameters as in Fig. 9. The labels of the symmetry points correspond to those for bcc BZ.

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In the pressure range 120

F. P at are lower.

In FIG. 11, in panel A, we have plotted the SH bond length as a function of the pressure and the amplitude of the calculated spread of this bond length due to the hydrogen zero point movement indicated by the red error bars. USPEX theory 12, 13 predicts the second order phase transition of

to R3m structure at 180 GPa, with the sulfur atoms remaining in the same places on the bcc unit cell, while hydrogen ions are frozen in one of the two minima of their double well potential due to the hydrogen bond as in ice structure transitions. On the contrary, this transition in the pressure range of 120-180 GPa, since the quantum zero point movement amplitude is forbidden, which is expected in the structure R3m to be greater than the difference between the short and long hydrogen bonds, therefore the ZPM stabilizes the structure in the pressure range 130-180 GPa^{3}in accordance with recent experiments

^{2}.

The top plate **(a)** shows the SH bond length in the pressure range between 120 and 180 GPa, where without the hydrogen ZPM the R3m structure with the SH bond splitting into a long (blue square) and short (violet squares) sulfur-hydrogen bond was expected to be stable. The amplitude of the calculated ZPM of the SH bond is indicated by the red error bars ^{3, 4} specified. The SH amplitude of the zero point movement, ZPM, is greater than the SH division in the range of 130–180 GPa, therefore the ZPM stabilizes the pressure range in this pressure range

^{4}in accordance with experiment

^{2}. The whiteboard (

**b**) shows the Fermi energy E

_{ F 4}in the small Fermi pocket at Γ as a functional print. The tip of this band crosses the chemical potential at 130 GPa of the Lifshitz Type 1 junction for the appearance of a new Fermi surface. The position of the VHS E

_{2}stays below the chemical potential, but stays in the energy range of mating interaction. blackboard

**(c)**shows the variation in experimental isotope coefficients calculated from data in ref. 2, which shows a deviation from 0.3 at 180 GPa to 1.5 at 135 GPa, which the BCS theory does not predict. The critical temperature drops towards zero with a decrease of about 60 K in a range of 30 GPa, beteen 160 GPa and 130 GPa, which were not predicted by the BCS theory. Both phenomena are identified by the general theory of superconductivity in the vicinity of a multi-distance Lifshitz transition

^{3, 4}predicted.

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Plate B in Fig. 11 shows the Fermi energy in the small hole Fermi pocket Γ as a function of pressure. The tip of this band crosses the chemical potential for a pressure greater than 130 GPa. At this pressure, the Type 1 Lifshitz transition to the appearance of a new Fermi surface occurs. In fact, over 130 GPa, a new little Fermi pocket appears at the Γ point of the Brillouin zone. The Fermi energy E _{ F 4} remains smaller than the energy from the pairing interaction cut below 160 GPa. Therefore, in the pressure range between 130 and 160 GPa, the Migdal approximation for the electron pairing in the small Fermi pocket at the Γ point of the Brillouin zone collapses.

Finally, panel c of Fig.11 the variation of the experimental isotope coefficient in this pressure range, calculated from recent data ^{2, } which show a regular increase with decreasing pressure from 0.3 at 180 GPa to 1.5 at 135 GPa. The divergence of the isotope coefficient of the Lifshitz transition at 130 GPa approximation is not predicted by the BCS theory using the standard Midgal approximation and an effective single band, but it is within the general theory of multi-distance superconductiivity near a Lifshitz transition ^{4} predicted. This is due to the decrease in the critical temperature of about 40 K in H. _{3} S and from about 60 K in D _{3} S, while the BCS calculations support the change in the critical temperature above this 30 GPa range, which is actually the critical temperature in the multigaps superconductors with a Lifshitz transition approaching zero (as in the Fano-Feshbach -Antiresonance), when a new nth Fermi surface occurs, in which electrons in the nth band have the energy E zero _{ F. } n = 0 at the band edge forming a BEC condensate, while the critical temperature has a maximum (Fano-Feshbach resonance) where the electrons have a Fermi energy of the order of the pairing energy forming a condensate in the BEC-BCS crossover ^{4} .

### Conclusions

In H _{3} S the onset and the maximum superconducting critical temperature, 203 K, are controlled by pressure, like in cuprates where the onset and the maximum value of the critical temperature 160 K is reached by tuning the lattice misfit strain at fixed doping ^{58} .

The calculated band structure for an ordered H _{3} S lattice as a function of pressure clearly shows multiple Lifshitz transitions for appearing of new Fermi surface spots in the pressure range showing high T _{ c } superconductivity, which together with quantum hydrogen zero point motion puts the system beyond the Migdal approximation.

New Fermi surface spots appear at the Γ point at 130 GPa pressure where the onset of the high critical temperature appears. It is possible that the appearing new Fermi surface spots drive the negative interference effect in the exchange interaction between multiple gaps ^{27} contributing to the suppression of the critical temperature ^{3} . This is supported by the isotope coefficient which diverges at 130 GPa reaching a value of 1.5 ^{3}, see Fig. 11, well beyond the predictions of single band Eliashberg theory. The divergence of the isotope coefficient observed here has already been observed in cuprates ^{59, 60} providing a clear experimental indication for an unconventional superconductivity near a Lifshitz transition ^{4} .

Increasing the pressure to 210 GPa a van Hove singularity crosses the chemical potential giving a Lifshitz transition for opening a neck. Moreover, the vhs remains near the chemical potential within the energy range of the energy cutoff for the pairing interaction over the full pressure range between 210 and 260 GPa. We show that the quantum zero point hydrogen fluctuations in a double well ^{3} typical of hydrogen bond and involving the T _{ u 2} phonon stretching mode, has strong effect on the electronic states near the Fermi level. The quantum hydrogen zero point motion, induces fluctuations of the 600 meV of the energy position of the vHs. The zero point amplitude of the SH stretching mode, involving the T _{ u 2} phonon, stabilizes the

Structure in the pressure range 130-180 GPa and induces large fluctuations of the small Fermi surface pockets at Γ. In conclusion we have shown the presence of large displacement amplitudes of ZPM. Single phonon waves can be disturbed by lattice quantum zero point disorder ^{45} but superconductivity seems to resist to perturbations from ZPM in H _{3} S. On the other hand, it is also seen that the DOS peak at E. _{ F. } seems to pass through E. _{ F. } with the large zero point motion. Finally more work is needed to investigate the variation of the Fermi level E. _{ F. } in different Fermi surfaces with different H isotopes which change the zero point motion amplitude.

### additional information

**How to cite this article** : Jarlborg, T. and Bianconi, A. Breakdown of the Migdal approximation at Lifshitz transitions with giant zero-point motion in H. _{3} S superconductor. Sci. Rep. **6**, 24816; doi: 10.1038 / srep24816 (2016).

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